(x-27)^3/2=64

3 min read Jun 17, 2024

Solving the Equation (x-27)^(3/2) = 64

This article will guide you through solving the equation (x-27)^(3/2) = 64. We'll break down the steps and explain the reasoning behind each one.

Before we start, let's understand what the fractional exponent (3/2) means:

The numerator (3) represents the power we need to raise the base to. In this case, we're cubing the base.
The denominator (2) represents the root we need to take. In this case, we're taking the square root.

So, (x-27)^(3/2) means we first cube (x-27) and then take the square root of the result.

Isolate the base: To get rid of the fractional exponent, we first need to isolate the base (x-27). We can do this by raising both sides of the equation to the power of (2/3), which is the reciprocal of (3/2).
```
[(x-27)^(3/2)]^(2/3) = 64^(2/3)
```
Simplify: The exponents cancel out on the left side, leaving us with (x-27). On the right side, we need to calculate 64^(2/3). This means taking the cube root of 64 (which is 4) and then squaring the result (which is 16).
```
x-27 = 16
```
Solve for x: Finally, we can solve for x by adding 27 to both sides of the equation:
```
x = 16 + 27
x = 43
```

Therefore, the solution to the equation (x-27)^(3/2) = 64 is x = 43.